How do you make sin(10) == sin(30) manipulating the bases of the numbers?

Question

Try as I might I can't figure out how to have those to sin functions equal each other. My professor suggested that I use a half angel identity but this hasn't helped me.

Answer 1

If you can have the two number be different bases,

just use base 30 on the left and base 10 on the right,

but that's probably not it.

I imagine he wants you to have the two numbers in the same base.

And we'll work in degrees rather than radians.

Sin(x) = sin(180 - x)

I'll write [n] to mean number in mystery base b,

but expressed in base 10 and {n} to mean number expressed in base b.

So we want [30] = [180 - 10].

or [10] = [180 - 30]

[10] is equal to the base itself and [30] = 3x base.

so 180 - b = 3b

180 = 4b

90 = 2b

45 = b

Base 45 ?

in base 45 {10} = 45 and {30} = 135

and sin of those two is the same.

yep.

Addendum:

with sin being a periodic function, other numbers will work too:

solutions to sin(3x) = sin(x):

45, 135, 180, 225, 315, 360 in the first revolution of the circle, then 360n + each of these.

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